Looks like a big fat bug of ESXi 6.5 U1 because I have the exact same issue here.
Brand new ESXi 6.5 U1 (Build 7388607), VMs with snapshots, can take new snapshot but cannot delete any.
VMDK chain is also broken and I also have two snapshot referencing the same delta files (*000001*) ....
And because I'm lucky, this is an Active Directory so it's a little bit harder to restore if I want to go down that path :-/
I tried removing SRVADPPRP-Snapshot2.vmsn and *00002* files then clear any reference to Snapshot2 in vmsd file but it looks like my tweaked vmsd file is not good enough because vsphere client does not display any snapshot when I load it.
Here is a ls -alh in VM's folder :
drwxr-xr-x 1 root root 80.0K Feb 21 14:12 .
drwxr-xr-t 1 root root 76.0K Feb 20 08:34 ..
-rw------- 1 root root 2.7G Feb 20 13:29 SRVADPPRP-000001-sesparse.vmdk
-rw------- 1 root root 414 Feb 20 13:30 SRVADPPRP-000001.vmdk
-rw------- 1 root root 249.0M Feb 21 08:52 SRVADPPRP-000002-sesparse.vmdk
-rw------- 1 root root 344 Feb 21 08:52 SRVADPPRP-000002.vmdk
-rw------- 1 root root 297.9M Feb 20 13:35 SRVADPPRP-000003-sesparse.vmdk
-rw------- 1 root root 367 Feb 20 13:35 SRVADPPRP-000003.vmdk
-rw------- 1 root root 1.2G Feb 21 14:28 SRVADPPRP-000004-sesparse.vmdk
-rw------- 1 root root 344 Feb 20 13:35 SRVADPPRP-000004.vmdk
-rw-r--r-- 1 root root 52 Feb 19 18:16 SRVADPPRP-964e1336.hlog
-rw------- 1 root root 8.0G Feb 19 18:26 SRVADPPRP-964e1336.vswp
-rw------- 1 root root 9.5K Feb 19 18:07 SRVADPPRP-Snapshot1.vmsn
-rw------- 1 root root 95.3K Feb 21 08:52 SRVADPPRP-Snapshot2.vmsn
-rw------- 1 root root 95.3K Feb 20 13:29 SRVADPPRP-Snapshot3.vmsn
-rw------- 1 root root 95.3K Feb 20 13:35 SRVADPPRP-Snapshot4.vmsn
-rw------- 1 root root 60.6G Feb 19 18:07 SRVADPPRP-flat.vmdk
-rw------- 1 root root 72.5K Feb 21 08:37 SRVADPPRP.nvram
-rw------- 1 root root 620 Feb 20 13:30 SRVADPPRP.vmdk
-rw-r--r-- 1 root root 1.8K Feb 21 08:52 SRVADPPRP.vmsd
-rwxr-xr-x 1 root root 3.5K Feb 20 13:35 SRVADPPRP.vmx
-rw------- 1 root root 0 Feb 19 18:26 SRVADPPRP.vmx.lck
-rw------- 1 root root 3.1K Feb 21 08:53 SRVADPPRP.vmxf
-rwxr-xr-x 1 root root 3.5K Feb 20 13:35 SRVADPPRP.vmx~
-rw------- 1 root root 441.3M Feb 20 13:29 SRVADPPRP_1-000001-sesparse.vmdk
-rw------- 1 root root 418 Feb 20 13:30 SRVADPPRP_1-000001.vmdk
-rw------- 1 root root 409.0M Feb 21 08:52 SRVADPPRP_1-000002-sesparse.vmdk
-rw------- 1 root root 348 Feb 21 08:52 SRVADPPRP_1-000002.vmdk
-rw------- 1 root root 425.0M Feb 20 13:35 SRVADPPRP_1-000003-sesparse.vmdk
-rw------- 1 root root 371 Feb 20 13:35 SRVADPPRP_1-000003.vmdk
-rw------- 1 root root 441.2M Feb 21 14:27 SRVADPPRP_1-000004-sesparse.vmdk
-rw------- 1 root root 348 Feb 20 13:35 SRVADPPRP_1-000004.vmdk
-rw------- 1 root root 100.0G Feb 19 18:06 SRVADPPRP_1-flat.vmdk
-rw------- 1 root root 624 Feb 20 13:30 SRVADPPRP_1.vmdk
Any help would be appreciated. Please note that I do have backups of my VM but I cannot power it off easily.